The curl of velocity, Kelvin's circulation theorem, and vortex dynamics
Vorticity is the fundamental measure of local rotation in a fluid. Mathematically, it is the curl of the velocity field. In two dimensions, the velocity field v = (u, v) has a single scalar vorticity component:
Positive vorticity means counterclockwise rotation; negative means clockwise. A key insight: fluid can travel in circles (like an irrotational vortex) without having any vorticity — the fluid elements translate around the origin but do not spin about their own centers. This distinction between circulation of the flow andlocal rotation of fluid elements is central to understanding vorticity.
The visualization below makes this tangible. A small paddle wheel follows your cursor and rotates at the local vorticity rate. Try the Irrotational Vortex preset and notice that the paddle does not spin, even though the fluid clearly swirls around the origin.
Uniform vorticity everywhere — the paddle wheel spins at a constant rate
Circulation is the line integral of velocity around a closed curve:
By Stokes' theorem, circulation equals the integral of vorticity over the area enclosed by the curve: Γ = &iint; ω dA. This means circulation around a contour that encloses a vortex core is non-zero, while a contour that avoids all vorticity gives Γ ≈ 0.
Click to place control points forming a closed polygon. The calculator numerically evaluates Γ by sampling the velocity at ~200 points along the contour. Drag the contour across the vortex center and watch the circulation jump.
One of the most powerful results in fluid dynamics: for an inviscid, barotropic fluid under conservative body forces, the circulation around any material loop (a loop that moves with the fluid) is constant in time:
This explains why vortices are so persistent in nature — once created, they cannot be destroyed in an ideal fluid. Viscosity is required to diffuse and ultimately dissipate vorticity. It also implies that in an initially irrotational flow, no vorticity can be generated internally (it must come from boundaries or non-conservative forces).
Point vortices are the simplest model of concentrated vorticity. Each vortex induces a velocity on every other vortex (but not itself) via the 2D Biot-Savart law: the induced velocity is tangential to the line connecting two vortices, with magnitude Γ/(2πr).
This leads to remarkable dynamics. Two like-signed vortices orbit each other. Two opposite-signed vortices of equal strength (a dipole) translate in a straight line. More complex configurations produce periodic leapfrogging or chaotic motion.
Two co-rotating vortices orbit their common center
Hermann von Helmholtz established three fundamental theorems about vortex motion in an ideal (inviscid) fluid (1858):
These theorems explain why smoke rings maintain their structure, why tornadoes are tube-shaped, and why trailing vortices from aircraft wings persist for miles behind the plane.